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Properties

Label 2-364-28.27-c1-0-27
Degree $2$
Conductor $364$
Sign $0.661 - 0.750i$
Analytic cond. $2.90655$
Root an. cond. $1.70486$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 1.32i)2-s + 2.64·3-s + (−1.50 − 1.32i)4-s + (−1.32 + 3.50i)6-s + 2.64·7-s + (2.50 − 1.32i)8-s + 4.00·9-s − 2.64i·11-s + (−3.96 − 3.50i)12-s − i·13-s + (−1.32 + 3.50i)14-s + (0.500 + 3.96i)16-s + (−2.00 + 5.29i)18-s − 5.29·19-s + 7.00·21-s + (3.50 + 1.32i)22-s + ⋯
L(s)  = 1  + (−0.353 + 0.935i)2-s + 1.52·3-s + (−0.750 − 0.661i)4-s + (−0.540 + 1.42i)6-s + 0.999·7-s + (0.883 − 0.467i)8-s + 1.33·9-s − 0.797i·11-s + (−1.14 − 1.01i)12-s − 0.277i·13-s + (−0.353 + 0.935i)14-s + (0.125 + 0.992i)16-s + (−0.471 + 1.24i)18-s − 1.21·19-s + 1.52·21-s + (0.746 + 0.282i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 - 0.750i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 364 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.661 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(364\)    =    \(2^{2} \cdot 7 \cdot 13\)
Sign: $0.661 - 0.750i$
Analytic conductor: \(2.90655\)
Root analytic conductor: \(1.70486\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{364} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 364,\ (\ :1/2),\ 0.661 - 0.750i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.66967 + 0.753717i\)
\(L(\frac12)\) \(\approx\) \(1.66967 + 0.753717i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 1.32i)T \)
7 \( 1 - 2.64T \)
13 \( 1 + iT \)
good3 \( 1 - 2.64T + 3T^{2} \)
5 \( 1 - 5T^{2} \)
11 \( 1 + 2.64iT - 11T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 5.29T + 19T^{2} \)
23 \( 1 - 7.93iT - 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + 2.64T + 31T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 - 7iT - 41T^{2} \)
43 \( 1 + 5.29iT - 43T^{2} \)
47 \( 1 + 7.93T + 47T^{2} \)
53 \( 1 - 4T + 53T^{2} \)
59 \( 1 + 10.5T + 59T^{2} \)
61 \( 1 - 7iT - 61T^{2} \)
67 \( 1 + 13.2iT - 67T^{2} \)
71 \( 1 - 15.8iT - 71T^{2} \)
73 \( 1 + 7iT - 73T^{2} \)
79 \( 1 + 13.2iT - 79T^{2} \)
83 \( 1 + 5.29T + 83T^{2} \)
89 \( 1 + 14iT - 89T^{2} \)
97 \( 1 + 7iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.30805395046880504813244198053, −10.35334785860889719890597355467, −9.224679860579102861636232742460, −8.612819428244674262057812784745, −7.974833284230237124559641151878, −7.20444887064878828040712720508, −5.82283381509617375380046704361, −4.61402973530226840842132696567, −3.39160677536440472396088069310, −1.69852090821316109022626954131, 1.79748004562692399511237543219, 2.61723776692082725441602347467, 4.00834549831180653712377574379, 4.78296993121953392227327505388, 6.99024399111233468039166660603, 8.039738074984828709027939630971, 8.600504301226130164398589540363, 9.315764092225712011077592039947, 10.36170768744278193085685606103, 11.09276228966929639915028359732

Graph of the $Z$-function along the critical line